Automorphism

In mathematics, an automorphism (from Greek auto, αὐτός and Greek μορφή morphé shape, form ) is an isomorphism of a mathematical object to itself

• 4.1 graphs 4.1.1 General
• 4.1.2 example

• 4.3.1 General
• 4.3.2 Inner automorphisms
• 4.3.3 Examples
• 4.3.4 Related Topics

From symmetries to automorphisms

An equilateral triangle has three axes of symmetry:

It also has a threefold rotational symmetry. To summarize the symmetry property mathematically, we consider the associated symmetry pictures. For each symmetry axis mirroring heard on the axis:

The numbers serve only to describe the picture, it is the same thing twice triangle. Symmetry figures may be carried out sequentially. In the following example, the sequential execution of two reflections is a rotation of 120 °:

If one carries twice the same mirroring one by one, you get a total of the picture that changed nothing, the identity mapping. If the sequential execution of two symmetry pictures should be a symmetry mapping again, so you must allow the identity map. A figure is unbalanced if it only allows this one, trivial symmetry mapping. The set of figures is a symmetry group, the symmetry group.

In mathematics one often considered objects, which consist of a basic amount and an additional structure, and usually there is a canonical structure that may result from the additional structure and a bijection a structure generated. In particular, it is possible for bijections.

Applied to the example corresponds to the symmetry plane and the triangle. For a congruence is the image triangle. Symmetry pictures are characterized by. In the abstract context are called bijections that satisfy automorphisms of. This definition covers most cases, be they graphs, topological spaces or algebraic structures such as vector spaces.

If the additional structures complicated the seemingly harmless condition can cause problems: If one defines differentiable manifolds as basic quantities with topology and an atlas, you might receive under a homeomorphism a compatible but not identical Atlas. If you were but in the definition require a maximum Atlas, would be such.

The category theory solves this and other problems by making it a pre-existing definition of structurally compatible pictures presupposes ( morphisms, it must not be actual figures). Based on this, it replaces the requirement of bijectivity (which is no longer in the abstract context is available ) by the existence of an inverse morphism.

Definition

Algebraic Structures

Be an algebraic structure, so a lot together with a (usually finite) number of links. Such an algebraic structure could for example be a vector space, a group or a ring. Then one sees in the algebra under a automorphism is a bijective mapping of the set onto itself, which is linear (or homomorphic ), ie it is

For everyone. The inverse function is automatically linearly in these circumstances.

Category theory

Be an object. A morphism is called an automorphism if it is a morphism

There is, it has a two-sided inverse.

An automorphism is the same order as

• An isomorphism, whose source and destination are the same, and
• An invertible endomorphism.

For categories of algebraic structures ( and the corresponding homomorphisms ) is equivalent to the definition in the previous section.

Automorphism

• The amount of all the automorphisms of an object together with the linkage as a linking group, which is referred to.
• If a group is called a homomorphism of a group operation of.
• Is a covariant functor and an object of so induces a group homomorphism. ( For contravariant functors one has to concatenate still with the inversion. ) Is given by a group operation on, we obtain in this way to an operation of.

Special Structures

Graphs

General

An automorphism of a graph with vertex set and edge set is a bijective mapping, such that for all.

An automorphism of a graph induces an automorphism of Komplementgraphen.

The set of fruit means that for each group there is a graph such that is isomorphic to.

Example

Be and:

Automorphisms of permutations are such that the application of the permutation again produces an illustration of the same graph on the chart. For example, the permutation being an automorphism, because the edges are still 1-2 and 3-4:

The permutation is not an automorphism, because the edges in the new image and are:

The automorphism group of the graph is isomorphic to the dihedral group of order, its complement is a 4 - cycle.

Vector spaces

An automorphism of a vector space is a bijective linear map.

For finite dimensional vector spaces automorphisms are precisely those linear transformations whose projection matrix is regular with respect to an arbitrary basis. The automorphism group is often listed as GL ( V).

Groups

General

An automorphism of a group is a bijective homomorphism, ie, a bijective mapping with for all.

Automorphisms get all the properties and structures that can be characterized with the aid of the group law. Examples: Every automorphism induces an automorphism of the center, receives the order of elements (ie, ) and forms a generating system on systems of generators from.

Inner automorphisms

If a group and firm, then, is an automorphism of, called conjugation. Automorphisms that arise in this way are called inner automorphisms. Automorphisms which are not inner automorphisms are called outer automorphisms. Because a homomorphism and exactly then is the trivial automorphism, if located in the center of the set of all inner automorphisms is isomorphic to a subgroup of. Indeed, it is a normal subgroup in, and the factor group is designated. They called the group of outer automorphisms. The restriction to the center provides a homomorphism.

For abelian groups, all internal homomorphisms are trivial, and.

For a subset is obtained by restricting the inner automorphisms an injective homomorphism. See normalizer and centralizer.

Examples

• The bijective mapping, is exactly then a homomorphism and thus an automorphism if it is Abelian.
• The group has exactly one non-trivial automorphism, viz. This follows from the fact that an automorphism maps a generating system to a system of generators.
• The automorphism group of small between group of four is isomorphic to the symmetric group.
• The automorphism group is (multiplication), the automorphism group is uncountable.
• The automorphism is not inner automorphism, because its restriction to the center, the subgroup of Skalarmatrizen, is not trivial.

Related Topics

• Subgroups that are invariant under all automorphisms are called characteristic subgroups.

Body

An automorphism of a body is a bijective mapping that and for all fulfilled. If a field extension, then called those automorphisms of the meet for all the automorphisms of. They form a group, listed or. An automorphism of an automorphism if and only if it is a linear map.

• For the conjugation of the body is an automorphism of the complex numbers.
• The figure is for the only non-trivial automorphism of.
• The field of rational numbers and the real numbers do not have non-trivial automorphisms. They are therefore referred to as rigid .. As the example shows, the rigidity is not transferred to bottom, top, intermediate body. That is rigid, indicated by the fact that every rational number can be as algebraic expression in pose, the must remain as neutral element of the multiplication under automorphisms. Each automorphism must map to corresponding each rational number to itself. Since he also receives the order must be even all real numbers fixed point.
• Is a finite or more generally a perfect field of characteristic, then is an automorphism of, the Frobenius.
• A body and a sub- quantity, is a subfield of, called the fixed field of. Is a finite subgroup, then is a Galois extension of degree. The Galois theory makes more connections between body extensions and automorphism groups.

Algebras

For algebras can be defined as groups of inner automorphisms as conjugation with a unit. Inner automorphisms are trivial on the center, and the set of Skolem -Noether states that the converse is true for a semisimple algebra.

Function theory

In the theory of functions holomorphic morphisms the functions and automorphisms are the conformal self-maps. The automorphism example of the open unit disk is given by: